1. Copyright © 2002 Thomson Learning, Inc.
Chapter 11 Appendix
Copyright © 2002 Thomson Learning, Inc.
Thomson Learning™ is a trademark used herein under license.
ALL RIGHTS RESERVED. Instructors of classes adopting PUBLIC FINANCE: A CONTEMPORARY APPLICATION OF THEORY TO POLICY, Seventh Edition by David N. Hyman as an assigned textbook may reproduce material from this publication for classroom use or in a secure electronic network environment that prevents downloading or reproducing the copyrighted material. Otherwise, no part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including, but not limited to, photocopying, recording, taping, Web distribution, information networks, or information storage and retrieval systems—without the written permission of the publisher.
Printed in the United States of America
ISBN X

2. Derivation of the Formula for Excess Burden of a Unit Tax
W = 1/2T × DQ
Where:
W = excess burden (Deadweight loss)
T = tax
DQ = change in equilibrium quantity as a result of the tax

3. Solving for DQ Step 1 Some Definitions T = PG – PN DPG = PG – P*
DPN = PN – P*

4. Solving for DQ Step 2 Elasticity comes into play ED =(DQ/Q*)/(DPG/P*)
=(DQ/Q*) × (P*/DPG)
=(DQ/Q*) × [P*/(PG – P*)]
ES =(DQ/Q*)/(DPN/P*)
=(DQ/Q*) × (P*/DPN)
= (DQ/Q*) × [P*/(PN – P*)]

5. Solving for DQ Step 3 Solving for PG ED = (DQ/Q*) × [P*/(PG – P*)]
(PG – P*) = (DQ/Q*) × (P*/ED) 
PG = (DQ/Q*) × (P*/ED) + P*

6. Solving for Q Step 4 Solving for PN ES = (DQ/Q*) × (P*/(PN – P*)
(PN – P*) = (DQ/Q*) × (P*/ES)
PN = (DQ/Q*) × (P*/ES) +P*

7. Solving for DQ Step 5 Using the T = PG – PN definition T = PG – PN
= (DQ/Q*) × (P*/ED) +P* – [(DQ/Q*) × (P*/ES) +P*]
= (DQ/Q*) × (P*/ED) – (DQ/Q*) × (P*/ES)
= (DQ/Q*) × (P*) × [(1/ED) – (1/ES)]
= (DQ/Q*) × (P*) × [(ES – ED)/(EDES)]

8. T = (DQ/Q*) × (P*) [(ES – ED)/(EDES)] for DQ
Solving for DQ
Step 6 Solving
T = (DQ/Q*) × (P*) [(ES – ED)/(EDES)] for DQ
T = (DQ/Q*) × (P*) [(ES – ED)/(EDES)] 
So 
DQ = T(P*/Q*) × [(EDES)/(ES – ED)] 

9. Plugging back into W = 1/2TDQ
W = 1/2T2(P*/Q*) × [(EDES)/(ES – ED)]

10. Derivation for the Ad-Valorem Tax
If the pre- and post-tax prices are close to one another, then
W = 1/2t2(P*Q*) × [(EDES) / (ES – ED)]
If LRAC is perfectly inelastic, then
W = 1/2t2 (P*Q*) × (ED) × [(ES)/(ES – ED)]
= 1/2t2 (P*Q*) × (ED)
because [(ES)/(ES – ED)] approaches 1.

11. Individual Losses in Welfare Under Perfect Competition
If there is perfect competition, then ED is infinite from the firm owner’s perspective.
This implies that
DWL = 1/2t2(P*Q*)ES

12. Compensated Demand Curves
Recall that compensated demand curves show the relationship between price and quantity demanded excluding the income effect. It only looks at the substitution away from the taxed good.

13. Figure 11A.1 Regular and Compensated Demand Curves For a Normal Good
Price
Curve
Regular Demand Q1
Gasoline per Year

14. Compensated Supply Curves
Recall that compensated supply curves show the relationship between price and quantity supplied excluding the income effect. It only looks at the substitution away from the taxed good.

15. ST DC DR S E2 PG Price (Dollars) 1.00 E1 A PN DQS Q2 Q1 Gasoline per
Figure 11A.2 Using A Compensated Demand Curve to Isolate The Substitution Effect of a Tax-Induced Price Increase ST DC DR S E2 PG
Price (Dollars)
1.00 E1 A PN
DQS Q2 Q1
Gasoline per
Year (Gallons) DQ

16. Figure 11A.3 A Compensated Supply Curve for an Input
Compensated Labor
Supply Curve W1
Wages
Regular Labor Supply Curve Q1
Input Services per Year